An example of a convex body without symmetric projections

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K such that for every k > C√n ln n and every projection P of rank k, the body PK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the "symmetric" theory to the general case.